Semiconductor metrology commonly requires measurement of periodic structures that are as large as or larger than the measurement spot of the measurement instrument. In some instances, these structures consist of test structures placed in the scribe line between dies of a semiconductor wafer. In other cases, these periodic structures are the structures forming the active circuits in the die (e.g., a memory array within a die). In yet other case, these periodic structures are small test structures placed within the die. In cases where the periodic structure substantially fills the measurement spot, it can generally be effectively modeled as an infinite structure.
During semiconductor wafer inspection, defects or particles commonly need to be detected on the device structures within the die. In many cases, these structures may be periodic or substantially periodic. For example, the structures may be periodic if they consist of a memory array, or if restrictive design rules require transistors to be laid out on a regular grid. In an inspection tool, a large area (such as a rectangle or line) may be illuminated simultaneously, whereby the illuminated area is then imaged onto a detector (e.g., CCD) such that different detector elements correspond to different locations within the illuminated area. If the image of a structure substantially fills one element of the detector and is substantially periodic, it may be sufficiently accurate to calculate its reflectivity by treating it as if it were an infinitely large periodic pattern.
The most commonly used method for calculating diffraction from a periodic structure on a semiconductor wafer is the rigorous coupled-wave analysis (RCWA), also referred to as the rigorous coupled-wave theory (RCWT). In general, RCWA computes diffraction by first dividing a given structure into a series of horizontal slabs. Then, within each slab, approximating the dielectric constant and the fields as finite sums of periodic functions of the horizontal position within the given slab. Commonly, the dielectric function (or the reciprocal of the dielectric function) is expressed as a finite Fourier series of position. Within each slab, the relationship between the fields at the top and bottom of the slab is calculated. Starting from the bottom, the relationships between fields at the top of one slab and the bottom of the next are computed, until the top of the structure is reached and the reflectivity of the structure as a whole can be computed given the specific illumination conditions.
Alternatively, other methods include those based on Green's functions and finite-difference time-domain (FDTD) methods. These methods are typically optimized for periodic structures when used in optical scatterometry metrology applications in the semiconductor industry.
However, when a finite sized structure or defect is combined with an effectively infinitely sized periodic structure, the resulting structure can no longer be regarded as an infinitely large periodic structure. Such situations are encountered commonly in the semiconductor industry. For example, a small metrology target may be placed on top of a large periodic pattern. The small target may be smaller than the measurement spot and so treating the structure as infinitely large may result in significant inaccuracies. Another common example includes settings where a defect exists within, on, or under a periodic pattern. In this setting, the defect may consist of extra material (e.g., a particle, bridge or oversized feature) or missing material (e.g., missing or undersized feature or a void). In this case, the structure is no longer periodic due to the extra or missing material that is present in only one unit cell and does not repeat (or is not at the same location) in other unit cells.
Traditionally, in order to approximately calculate the reflectivity, diffraction or scattering of such non-periodic structures using an algorithm, such as RCWA that assumes a periodic structure, it is necessary to construct a new larger unit cell containing multiple unit cells of the underlying structure. In the case of a small defect, only one of those unit cells actually contains the defect. In the case of a small target on top of a larger grating, the new unit cell must contain the entire small target. In order to obtain a reasonably accurate result the new unit cell should be large enough that the electric fields from the non-repeating feature have decayed to a relatively small value at the edge of the new periodic structure. For example, in the case of a small defect, this may require that the unit cell be a few (approximately 3-5) wavelengths in each dimension. By way of another example, in the case of the small metrology target, the new unit cell may need to be several times the size of the small target.
By way of example, in the case of the defect, the pitch of the original periodic structure might be less than 100 nm (e.g., 30-50 nm), in one direction in a dense memory array. If the longest illumination wavelength is approximately 300 nm, then the larger unit cell that includes the defect would need to be ten or more times larger than the period of the repeating structure. In the case of a 2D grating, the pitch in the other repeating direction would be even larger (e.g., 300-500 nm). As such, along that direction increasing the unit cell size two to three times may be sufficient. In this case, the example given the unit cell may require a size that is 10-30 times the size of the periodic structure. In order to resolve the same small features, the truncation order has to be increased by the same factor in each dimension as the unit cell increases the given dimension. Computation times for RCWA typically scale as the cube of the truncation order (for large truncation orders). As such, the computation time required to account for such a feature might increase by a factor of 1000 or more if the accuracy of the computation is to be maintained. Further, the memory requirements typically scale approximately as the square of the truncation order, requiring a factor of 100 times more memory for a ten-fold increase in the truncation order. Therefore, it would be desirable to provide a system and method that cures the defects of the prior.